A speaks truth in 80% of the cases while B speaks truth in 60% of the cases. Find the probability that they will contradict each other while stating the same fact.

Answer

To solve the problem, we need to find the probability that A and B will contradict each other when stating the same fact.

### Step-by-Step Solution:

1. **Identify the probabilities:**
- Probability that A speaks the truth, \( P(A) = 0.8 \) (or \( \frac{80}{100} \)).
- Probability that B speaks the truth, \( P(B) = 0.6 \) (or \( \frac{60}{100} \)).

2. **Calculate the probabilities of them not speaking the truth:**
- Probability that A does not speak the truth, \( P(A') = 1 - P(A) = 1 - 0.8 = 0.2 \) (or \( \frac{20}{100} \)).
- Probability that B does not speak the truth, \( P(B') = 1 - P(B) = 1 - 0.6 = 0.4 \) (or \( \frac{40}{100} \)).

3. **Define the cases of contradiction:**
- Case 1: A speaks the truth and B does not speak the truth.
- Case 2: A does not speak the truth and B speaks the truth.

4. **Calculate the probabilities for each case:**
- For Case 1:
\[ P(A \text{ speaks truth}) \times P(B' \text{ does not speak truth}) = P(A) \times P(B') = 0.8 \times 0.4 = 0.32 \]
- For Case 2:
\[ P(A' \text{ does not speak truth}) \times P(B \text{ speaks truth}) = P(A') \times P(B) = 0.2 \times 0.6 = 0.12 \]

5. **Add the probabilities of both cases to find the total probability of contradiction:**
\[ P(\text{Contradiction}) = P(A \text{ speaks truth}) \times P(B') + P(A') \times P(B) = 0.32 + 0.12 = 0.44 \]

6. **Express the total probability in fractional form:**
\[ P(\text{Contradiction}) = 0.44 = \frac{44}{100} = \frac{11}{25} \]

### Final Answer:
The probability that A and B will contradict each other while stating the same fact is \( \frac{11}{25} \).